Bifurcation Analysis of an Inverted Pendulum with Saturated Hamiltonian Control Laws

2000 ◽  
Vol 33 (2) ◽  
pp. 173-174
Author(s):  
Enrique Ponce ◽  
Javier Aracil ◽  
Francisco Salas ◽  
Daniel Juan Pagano
Keyword(s):  
Bifurcation Analysis ◽  
Inverted Pendulum ◽  
Control Laws

scholarly journals A reversible bifurcation analysis of the inverted pendulum

1998 ◽  
Vol 112 (1-2) ◽  
pp. 50-63 ◽  
Author(s):  
H.W. Broer ◽  
I. Hoveijn ◽  
M. van Noort
Keyword(s):  
Bifurcation Analysis ◽  
Inverted Pendulum

scholarly journals On the Dynamics of the Furuta Pendulum

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Benjamin Seth Cazzolato ◽  
Zebb Prime
Keyword(s):  
Nonlinear Control ◽  
System Dynamics ◽  
Inverted Pendulum ◽  
Lagrangian Formulation ◽  
Full System ◽  
Control Laws ◽  
Control Community

The Furuta pendulum, or rotational inverted pendulum, is a system found in many control labs. It provides a compact yet impressive platform for control demonstrations and draws the attention of the control community as a platform for the development of nonlinear control laws. Despite the popularity of the platform, there are very few papers which employ the correct dynamics and only one that derives the full system dynamics. In this paper, the full dynamics of the Furuta pendulum are derived using two methods: a Lagrangian formulation and an iterative Newton-Euler formulation. Approximations are made to the full dynamics which converge to the more commonly presented expressions. The system dynamics are then linearised using a Jacobian. To illustrate the influence the commonly neglected inertia terms have on the system dynamics, a brief example is offered.

Continuum of Motion Equations and Control Laws for the Inverted Pendulum Cart and Rotary Pendulum

2020 ◽  
Author(s):  
Constance Lare ◽  
Warren N. White
Keyword(s):  
Inverted Pendulum ◽  
Equations Of Motion ◽  
Control Law ◽  
Control Laws ◽  
Motion Equations ◽  
Base Radius ◽  
Limiting Condition ◽  
And Control ◽  
The Given ◽  
Insight Into

Abstract This paper questions whether the controller properties for a given rigid body mechanical system still apply as the given system is changed. As a first attempt in this investigation, the controller for the underactuated rotary pendulum is investigated as the system morphs into an underactuated inverted pendulum cart. As the limiting condition of the inverted pendulum cart is approached, the investigation allows the controller to also morph. The authors show that, as the pendulum base radius grows, the rotary pendulum equations of motion morph into the inverted pendulum cart dynamics. The paper presents necessary conditions for the successful morphing of the dynamic equations. The morphing process for the controller tests the idea whether the control law also satisfies the same continuum basis as the motion equations. The paper presents a framework for the class of controllers investigated for providing insight into when the controller morphing may be successful. This paper presents dimensionless quantities that render the equations of motion and controller for the inverted pendulum cart and rotary pendulum into dimensionless form. These dimensionless quantities allow comparison of controllers and systems that are not possible through simple inspection. This comparison ability is especially useful for quantifying the nonlinearities of a given system and controller compared to another system and controller having different parameter sizes, a comparison rarely seen in the control literature.

BIFURCATION ANALYSIS OF A ROTATING ARM WITH SATURATED HAMILTONIAN CONTROL LAWS

2005 ◽  
Vol 15 (10) ◽  
pp. 3223-3243
Author(s):  
F. SALAS ◽  
E. PONCE ◽  
J. ARACIL ◽  
F. GORDILLO
Keyword(s):  
Bifurcation Analysis ◽  
Actuator Saturation ◽  
Space Structure ◽  
Control Law ◽  
Control Laws ◽  
Codimension Two ◽  
Bifurcation Points ◽  
Dynamical Complexity ◽  
Saturation Effects ◽  
Codimension Two Bifurcation

The actuator saturation effects on the global state space structure of a rotating arm with a Hamiltonian control law are studied via bifurcation analysis. In addition to other well-known codimension-two bifurcation points, some saturation-induced "fan-like" bifurcation points are detected. The dynamical complexity found in this seemingly simple example is remarkable.

BIFURCATION ANALYSIS AND CONTROL OF NONLINEAR SYSTEMS WITH A NONSEMISIMPLE ZERO AT CRITICALITY AND APPLICATION

2000 ◽  
Vol 10 (08) ◽  
pp. 1887-1901
Author(s):  
JYUN-HORNG (ALEX) FU
Keyword(s):  
Nonlinear Systems ◽  
Bifurcation Analysis ◽  
Inverted Pendulum ◽  
And Control

Bifurcation analysis and control of nonlinear systems with defective repetitive zero at criticality is presented. The results are applied to stabilize a flexible, inverted pendulum mounted on a moving cart that is approximated with a coefficient of flexibility.

scholarly journals Bifurcation Analysis and Dynamic Behaviour of an Inverted Pendulum with Bounded Control

2016 ◽  
Vol 46 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Svetoslav Nikolov ◽  
Valentin Nedev
Keyword(s):  
Dynamic System ◽  
Bifurcation Analysis ◽  
Inverted Pendulum ◽  
Dynamic Behaviour ◽  
Nonlinear Differential Equations ◽  
Analytical Formula ◽  
Bounded Control ◽  
Pivot Point ◽  
Bifurcation Behaviour ◽  
Boundary Of Stability

Abstract This paper presents an investigation on the behaviour of con- ventional inverted pendulum with an inertia disk in its free extreme. The system is actuated by means of torques applied to the disk by a DC mo- tor, mounted on the pendulum’s arm. Thus, the system is underactuated since the pendulum can rotate freely around its pivot point. The dynam- ical model is given with three ordinary nonlinear differential equations. Using Poincare-Andronov-Hopf’s theory, we find a new analytical formula for the first Lyapunov’s value at the boundary of stability. It enables one to study in detail the bifurcation behaviour of the above dynamic system. We check the validity of our analytical results on the first Lyapunov’s value by numerical simulations. Hence, we find some new results.

scholarly journals Stabilizing a Rotary Inverted Pendulum Based on Logarithmic Lyapunov Function

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Jie Wen ◽  
Yuanhao Shi ◽  
Xiaonong Lu
Keyword(s):  
Lyapunov Function ◽  
Inverted Pendulum ◽  
Control Method ◽  
Quadratic Function ◽  
Control Law ◽  
Logarithmic Function ◽  
Control Laws ◽  
Lyapunov Control ◽  
Comparative Results ◽  
Rotary Inverted Pendulum

The stabilization of a Rotary Inverted Pendulum based on Lyapunov stability theorem is investigated in this paper. The key of designing control laws by Lyapunov control method is the construction of Lyapunov function. A logarithmic function is constructed as the Lyapunov function and is compared with the usual quadratic function theoretically. The comparative results show that the constructed logarithmic function has higher numerical accuracy and faster convergence speed than the usual quadratic function. On this basis, the control law of stabilizing Rotary Inverted Pendulum is designed based on the constructed logarithmic function by Lyapunov control method. The effectiveness of the designed control law is verified by experiments and is compared with LQR controller and the control law designed based on the quadratic function. Moreover, the system robustness is analyzed when the system parameters contain uncertainties under the designed control law.

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